>[!quote] *A personification* of a [[../../02 Areas/Math/Augmented Matrix#Row Operations|Row Operation]]. An [[Elementary Matrix]] is a square [[Matrix]] that represents a [[Augmented Matrix|Row Operation]] being done to an [[Augmented Matrix]]. Every [[Augmented Matrix|Row Operation]] has a corresponding [[Augmented Matrix|Row Operation]]. When representing an [[Augmented Matrix]] as $A\vec x = \vec b$, you can more rigorously denote row operations as multiplying both sides of the equations by an [[Elementary Matrix]]. >[!example] Add $2$ times the first row to the second row. >$\begin{align} >\mat{1 & 0 &0 \\ 2& 1 &0\\0&0&1} >\end{align}$ | [[Augmented Matrix\|Row Operation]] | [[Elementary Matrix]] | [[Determinant]] | | ---------------------------------------- | ---------------------------------------------------------- | --------------- | | Add a multiple $k$ of row $i$ to row $j$ | Take the [[Identity Matrix]] and replace $A_{ij}$ with $k$ | $\det A = 1$ | | Switch row $i$ and row $j$ | | $\det A=-1$ | | Scaling row $i$ by [[Scalar]] $m$ | | $\det A = m$ | A personification of [[Augmented Matrix]] Row